It is known that smooth closed oriented 4-manifolds are smooth branched covers of the 4-sphere. In this talk we extend this result to open 4-manifolds, by showing that they are branched covers of suitable open subsets of S^4. This has two main consequences: (1) any exotic R^4 is a smooth branched cover of the standard R^4, and (2) any closed oriented topological 4-manifold is a topological branched cover of S^4 (in the sense of Ralph Fox). Unfortunately, the branching set tends to be very wild at infinity. This is a joint work with Riccardo Piergallini (University of Camerino).